Frequency estimation systems are an important subsystem of communication, navigation, radar and various other engineering systems. In some cases, the ability to perform relatively efficient and precise estimation of frequency may be a critical component in system design. Indeed, an inability to perform relatively efficient and precise estimation may significantly limit the performance of these systems as measured by various metrics of performance. For example, in high dynamic global positioning satellite (GPS) receiver applications, effectively acquiring and tracking the GPS carrier signal under dynamic conditions may limit the performance and applicability of these receivers to various applications. For this reason, various architectures suited to such applications have been previously proposed that acquire and track the GPS carrier under dynamic conditions, with the various different methods having a variety of limitations in terms of signal-to-noise ratio, initial frequency uncertainty, and other metrics of performance.
In terms of communication systems, precise frequency and phase may be important in communication systems involving coherent modulation techniques such as multiple quadrature amplitude modulation (MQAM) and multiple phase shift keying (MPSK). In certain communication applications, techniques, such as square law loop or Costas type loop, are used to derive the carrier frequency and phase from modulated signals, or a pilot signal is used which is tracked. The use of square law loop or Costas type loop result in significant loss in terms of phase noise of the reference carrier and phase ambiguity problems associated with phase ambiguity in the carrier phase equal to integer multiple of 2π/M for the MPSK signal. The use of pilot carrier results in a loss of signal power because a significant part of available power is used up in the pilot. The ability to provide fast and accurate frequency and phase estimates at very low signal-to-noise ratios (SNRs) may reduce the loss due to pilot carrier to an insignificant value.
More recently, precise and fast frequency acquisition and tracking have become increasingly important with the evolution of the Orthogonal Frequency Division Multiplexing (OFDM) in mobile communication systems. The OFDM modulation scheme may reduce problems of inter symbol interference (ISI) caused by multipath propagation. It may also exhibit relatively high performance in selective fading environments. Due to these and other features, OFDM has become part of various standards such as Worldwide Interoperability for Microwave Access (WiMax). Because OFDM is based on the orthogonality among various subcarrier signals, it is very important that this orthogonality be maintained when these subcarriers are received at the receiver. However, the mobile wireless channels introduce frequency offsets which cause disruption of the orthogonality among the subcarriers resulting in mutual interference among the various subcarriers. Therefore, it is desirable to precisely estimate such frequency offsets and correct them to avoid problems associated with intercarrier interference. The offsets may be functions of time and may vary with different subcarriers. Therefore, it is further desirable that precise estimation of the frequency offset be made with a minimum requirement on the estimation time and SNR, which is also limited in systems involving error correction coding techniques.
Some of the previous techniques for frequency estimation are based on the extended Kalman filter (EKF). In these techniques, the state of the Kalman filter is comprised of the signal phase, frequency and possibly one or more derivatives of the frequency, and the measurement is a nonlinear function of the state vector. In this approach, the measurement function is linearized about the current estimate of the state. Thus the technique based on EKF is more appropriate in the tracking mode when the initial estimate of the state is close to the true state and under a relatively high signal-to-noise ratio condition. However, when fast acquisition is required starting with a relatively high frequency uncertainty and/or in low signal-to-noise ratio conditions, the EKF based methods may not meet the required performance as they have relatively high thresholds for required signal power to noise density ratio (P/N0). Therefore, if the P/N0 ratio is below the threshold, the EKF may fail to converge and instead it may diverge. In other words, when the EKF diverges, the state estimation error, instead of converging to relatively small value as more and more measurement samples are processed, increases with the number of samples processed and approaches a large estimation error. The EKF estimator also results in a phase locked loop (PLL) configuration with time-varying loop filter coefficients and is thus also an “optimum” PLL. Thus the PLLs have similar limitations as the EKFs.
In fast Fourier transform (FFT) methods for frequency estimation, the N-point Fourier transform of a signal is evaluated, and the peak of the absolute values of the FFT is searched. The FFT frequency corresponding to the peak is taken as the estimate of the unknown frequency associated with the signal. The FFT method has two limitations. First, the estimation error is of the order of the FFT resolution frequency that is equal to (2 B/N) where 2 B is the interval of frequency uncertainty. Second, in a relatively low SNR condition, selection of the peak among N noisy outputs of the transform involves detection errors corresponding to a large estimation error in the range of −B to B Hz. Increasing the sampling (N) reduces the resolution frequency; however, concurrently it also increases the probability of detecting the incorrect peak thus exhibiting threshold effects with respect to the SNR level.
The techniques that have been used in the estimation of the frequency offset in OFDM systems, based on the assumption that the offset is same for all the sub-carriers, involve correlating the received signal with a reference signal and thereby determining the relative phase between the two signals. By estimating the relative phase at two different time instances and dividing the difference between the two relative phase differences by the time difference provides the frequency estimate. The correlation operation requires that the integration interval TI for the correlation operation may be much smaller than the inverse of the offset frequency fa else the signal amplitude at the correlator output will be small and close to zero if TI≅1/fa. On the other hand, selection of a small TI leads to relatively large noise at the integrator output which may result in a noisy phase estimate. Differencing the noisy phase estimates may further accentuate the estimation error. Thus the correlation based approach has limitations in terms of the magnitude of the frequency offset and the SNR condition under which it will provide a satisfactory result.